One of my favorite explanations is this one from Ask A Mathematician. Basically starting off of the idea of finding the volume of the parallelogram that any transformation takes the unit cube, it creates from scratch the definition of determinant. I found this extremely amazing because usually determinants are introduced in the "here's how you calculate it" way and then later shown to be the volume of the transformed unit cube, but this resource starts from the very intuitive notion of "volume of the transformed unit cube" and builds the formula from there, based on some very important but still intuitive properties. Brilliant!
This "volume of transformed unit cube" idea is why it's important in inconsistent systems and invertible transformations/matrices; if the determinant is $0$ (i.e it takes the unit cube and squishes it into a lower dimension), then that means that the transformation is not invertible (not injective) and hence may not have solutions or has infinitely many solutions.
For the cross product, this resource may be illuminating: Motivation for construction of cross-product (Quaternions?). The links therein of course also provide more information, but I personally really like the one I explicitly linked. I'm not entirely sure why determinants are involved, but understanding the formula for the cross product may help in that quest for understanding. If someone has ideas, please comment or answer! I would love to know too.
And finally for eigenvalues, we want to find vectors and eigenvalues such that $A\vec v = \lambda \vec v$, which is equivalent to finding vectors in the null space of $(A-\lambda I)$. Like I said in part $(b)$, determinants are really good at telling us if the unit cube has been squished to a lower dimension (i.e. a non-zero vector has been sent by the transformation to $0$), so it's good at telling us what values of $\lambda$ give us matrices that have a non-zero null space, and hence telling us the eigenvectors/eigenspaces corresponding to a particular $\lambda$.
I might be adding more details later, but if you have questions on what I've written so far, feel free to ask.
P.S. if you want a more formal introduction to determinants and eigenvalues, one of my favorite linear algebra books has been Linear Algebra Done Right, by Sheldon Axler. See chapter 10 for determinants and chapter 5 for eigenvalues. It really sheds light on alternate ways of thinking about these things (outside of the common approach of "here's a crap ton of matrices, do some computations with them")
P.P.S I also wonder a lot about these "origin" questions in mathematics, and if you're interested in more, I'll share some awesome stuff that I've found over eons of prowling MSE:
The answers in all the things I've linked above are just so helpful for any student of mathematics I just couldn't resist sharing!